Vectors vs kets In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer almost exclusively to quantities like
displacement or
velocity, which have components that relate directly to the three dimensions of
space, or relativistically, to the four of
spacetime. Such vectors are typically denoted with over arrows (\vec r), boldface (\mathbf{p}) or indices (v^\mu). In quantum mechanics, a quantum state is typically represented as an element of a complex
Hilbert space, for example, the infinite-dimensional vector space of all possible
wavefunctions (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically. To distinguish this type of vector from those described above, it is common and useful in physics to denote an element \phi of an abstract complex vector space as a ket |\phi\rangle, to refer to it as a "ket" rather than as a vector, and to pronounce it "ket-\phi" or "ket-A" for . Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the |\ \rangle making clear that the label indicates a vector in vector space. In other words, the symbol "" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "" by itself does not. For example, is not necessarily equal to . Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling
energy eigenkets in quantum mechanics through a listing of their
quantum numbers. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as \hat x, \hat p, \hat L_z, etc.
Notation Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example: :\begin{align} \end{align} Note how the last line above involves infinitely many different kets, one for each real number . Since the ket is an element of a vector space, a
bra \langle A| is an element of its
dual space, i.e. a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts. A bra \langle\phi| and a ket |\psi\rangle (i.e. a functional and a vector), can be combined to an operator |\psi\rangle\langle\phi| of rank one with
outer product :|\psi\rangle\langle\phi| \colon |\xi\rangle \mapsto |\psi\rangle\langle\phi|\xi\rangle ~.
Inner product and bra–ket identification on Hilbert space Bra–ket notation is particularly useful in Hilbert spaces which have an inner product that allows
Hermitian conjugation and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (see
Riesz representation theorem). The inner product on Hilbert space (\ , \ ) (with the first argument anti linear as preferred by physicists) is fully equivalent to an (anti-linear) identification between the space of kets and that of bras in bra–ket notation: for a vector ket \psi = |\psi\rangle define a functional (i.e. bra) f_\phi = \langle\phi| by :(\phi,\psi) = (|\phi\rangle, |\psi\rangle) =: f_\phi(\psi) = \langle\phi| \, \bigl(|\psi\rangle\bigr) =: \langle\phi{\mid}\psi\rangle
Bras and kets as row and column vectors In the simple case where we consider the vector space \Complex^n, a ket can be identified with a
column vector, and a bra as a
row vector. If, moreover, we use the standard Hermitian inner product on \Complex^n, the bra corresponding to a ket, in particular a bra and a ket with the same label are
conjugate transpose. Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply imply
matrix multiplication. In particular the outer product |\psi\rangle\langle\phi| of a column and a row vector ket and bra can be identified with matrix multiplication (column vector times row vector equals matrix). For a finite-dimensional vector space, using a fixed
orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector: \langle A | B \rangle \doteq A_1^* B_1 + A_2^* B_2 + \cdots + A_N^* B_N = \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \begin{pmatrix} B_1 \\ B_2 \\ \vdots \\ B_N \end{pmatrix} Based on this, the bras and kets can be defined as: \begin{align} \langle A | &\doteq \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \\ \end{align} and then it is understood that a bra next to a ket implies matrix multiplication. The
conjugate transpose (also called
Hermitian conjugate) of a bra is the corresponding ket and vice versa: \langle A |^\dagger = |A \rangle, \quad |A \rangle^\dagger = \langle A | because if one starts with the bra \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \,, then performs a
complex conjugation, and then a
matrix transpose, one ends up with the ket \begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_N \end{pmatrix} Writing elements of a finite dimensional (or
mutatis mutandis, countably infinite) vector space as a column vector of numbers requires picking a
basis. Picking a basis is not always helpful because quantum mechanics calculations involve frequently switching between different bases (e.g. position basis, momentum basis, energy eigenbasis), and one can write something like "" without committing to any particular basis. In situations involving two different important basis vectors, the basis vectors can be taken in the notation explicitly and here will be referred simply as "" and "".
Non-normalizable states and non-Hilbert spaces Bra–ket notation can be used even if the vector space is not a
Hilbert space. In quantum mechanics, it is common practice to write down kets which have infinite
norm, i.e. non-
normalizable wavefunctions. Examples include states whose wavefunctions are
Dirac delta functions or infinite
plane waves. These do not, technically, belong to the
Hilbert space itself. However, the definition of "Hilbert space" can be broadened to accommodate these states (see the
Gelfand–Naimark–Segal construction or
rigged Hilbert spaces). Bra–ket notation continues to work in an analogous way in this more general context.
Banach spaces are a different generalization of Hilbert spaces. In a Banach space , the vectors may be notated by kets and the continuous
linear functionals by bras. Over any vector space without a given
topology, we may still notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the
Riesz representation theorem does not apply. ==Usage in quantum mechanics==